# Questions tagged [optimization]

This tag is intended for questions on methods for the (constrained or unconstrained) minimization or maximization of functions.

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### minimization problem: sum of Rayleigh quotients

I would like to find $x$ which minimizes the following equation: $\frac{x^HAx}{x^HBx} + \frac{x^HCx}{x^HDx}$ where A, B, C, D are positive-definite. $x$ is not a very large vector (<1000 elements ...
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### Maximization variant of semidefinite programming (SDP)

Consider the following program: $$\max_{\pmb a} \sum_i z_i\\ u.c. \pmb a \pmb P_i\pmb a^\top\geq z_i$$ where $\pmb a \in\mathbb{R}^p$ and the $\pmb P_i$ are all symmetric positive semidefinite ...
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In these notes (section 2.3), it is stated that: A point $x^*$ is a minimizer of a function $f$ (not necessarily convex) if and only if $f$ is subdifferentiable at $x^*$ and $0 \in\partial f(x^*).$ ...
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I have read this and other threads on this site on BFGS, but I still don't have a clear understanding of what it's meant by low-rank updates. For example, I read the following in this book: The ...
175 views

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### How to avoid the round-off errors in the larger calculations?

Now I need to sum up more than one thousands of terms and then make the four-dimmensional integral in my Fortran program. I found that there are some numerical errors. Can you give me some suggestions ...
392 views

### What method do you suggest to solve this minimax, quadratic in both variables problem?

I have a problem of the form, \begin{align} minimize_{y} maximize_{x}&\quad x^T y - y^T (B\odot x x^T) y\\ s.t. &x\in [l,u]\\ &Ay=b \end{align} How to efficiently solve this problem? ...
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### How to define the derivative for Scipy.Optimize.Minimize

I am trying to use scipy.optimize.minimize to minimise a quadratic objective function: $f(x) =x^\top Q x$. As a start, I have successfully implemented this using the built-in Nelder-Mead Simplex ...
412 views

### Feasibility checking

Consider the following optimization problem: $Min\;\;\; CX$ $AX\geq b$ $x_ix_j= x_s x_t\;\;\; i\neq j \neq s\neq t$ $x_j\geq 0;$ Where $A$ is the adjacency matrix and $C$ is a constant vector. ...
177 views

### Optimization of a blackbox function

Let's say that we have an objective function $f(\mathbf x,\mathbf y)$ which has the parameters $\mathbf x=[x_1\ldots x_n]$ and $\mathbf y=[y_1\ldots y_n]$. Here, $\mathbf y$ is a blackbox variable ...
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### Minimizing the ratio of two specific non negative quadratic convex functions

$F$ is $m\times m$ diagonal, with real non negative elements $D$ is $n \times m$ complex $P$ is $n \times 1$ complex $A$ is $m \times 1$ complex. Minimize $\Gamma(A)$, with respect to $A$. \...